Quantum Computer Jaewan Kim jaewan@kias.re.kr School of - - PowerPoint PPT Presentation

Quantum Computer

Jaewan Kim jaewan@kias.re.kr School of Computational Sciences Korea Institute for Advanced Study

KIAS (Korea Institute for Advanced Study)

• Established in 1996
• Located in Seoul, Korea
• Pure Theoretical Basic Science
• Schools

– Mathematics – Physics – Computational Sciences

• By the computation, For the computation
• 20 Prof’s, 2 Distinguished Prof’s, 20 KIAS Scholars,

70 Research Fellows, 20 Staff Members

Quantum Physics Information Science Quantum Parallelism Quantum computing is

exponentially larger and faster than digital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Body Simulation (Nanotechnology)

No Clonability of Quantum Information, Irrevesability of Quantum Measurement Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement Quantum Teleportation, Quantum Superdense Coding,

Quantum Cryptography, Quantum Imaging

Quantum Information Science

Twenty Questions

• Animal …
• Four legs?
• Herbivorous?

. . .

• Tiger?

Yes(1) or No(0) Yes(1) or No(0) Yes(1) or No(0) . .

It from Bit

John A. Wheeler

x x x x x x x x x

1 1 1

Yoot = Ancient Korean Binary Dice

1

Qubit

1 a b j = +

Taegeukgi (Korean National Flag)

Qubit = quantum bit (binary digit)

Businessweek 21 Ideas for the 21st Century

Transition to Quantum

• Mathematics: Real Complex
• Physics: Classical Quantum
• Digital Information Processing

– Hardwares by Quantum Mechanics – Softwares and Operating Systems by Classical Ideas

• Quantum Information Processing

– All by Quantum Ideas

• Quantum Computer

– Quantum Algorithms: Softwares

• Simulation of quantum many-body systems
• Factoring large integers
• Database search

– Experiments: Hardwares

• Ion Traps
• NMR
• Cavity QED, etc.

BT NT IT

Quantum Information Processing

• Quantum Communication

– Quantum Cryptography

• Absolutely secure digital communication
• Generation and Distribution of Quantum Key

– ~100 km through optical fiber (Toshiba) – 23.4 km wireless secure satellite communication

– Quantum Teleportation

• Photons
• Atoms, Molecules
• Quantum Imaging and Quantum Metrology

IT

Quantum Information Processing

Cryptography and QIP

• Public Key Cryptosystem (Asymmetric)

– Computationally Secure – Based on unproven mathematical conjectures – Cursed by Quantum Computation

• One-Time Pad (Symmetric)

– Unconditionally Secure – Impractical – Saved by Quantum Cryptography

• 병 주고 약 주고-

Giving disease, Giving medicine. Out with the old, In with the new.

KIAS-ETRI, December 2005. 25 km Quantum Cryptography T.G. Noh and JK

Classical Computation

• Hilbert (1900): 23 most challenging math problems

– The Last One: Is there a mechanical

procedure by which the truth or falsity of any mathematical conjecture could be decided?

• Turing

– Conjecture ~ Sequence of 0’s and 1’s – Read/Write Head: Logic Gates – Model of Modern Computers

Turing Machine

q′

q

' s s →

d

( , ) ( , ) ( , ) q f q s s g q s d d q s ′ =   ′ =   = 

Finite State Machine: Head Infinitely long tape: Storage

Bit {0,1

}

Bit and Logic Gate

“Universal” “Reversible” “Universal”

NAND C-NOT CCN ( Toffoli ) C-Exch ( Fredkin ) NOT

01 10

AND

000 010 100 111

OR

000 011 101 111

XOR

000 011 101 110

“Reversible”

DNA Computing

• Bit { 0, 1 } Tetrit (?) { A, G, T, C}
• Gate Enzyme
• Parallel Ensemble Computation

– Hamiltonian Path Problem

Adleman

N= (# bits to describe the problem, size of the problem) (#steps to solve the problem) = Pol(N) “P(polynomial)”: Tractable, easy (#steps to verify the solution) = Pol(N) “NP (nondeterministic polynomial)”: Intractable

Complexity

P NP

NP⊃P NP≠P or NP=P?

Quantum Information

• Bit
• Qubit
• N bits 2N states, One at a time

Linearly parallel computing AT BEST

• N qubits Linear superposition
• f 2N states at the same time

Exponentially parallel computing Quantum Parallelism But when you extract result, you cannot get all of them.

2 2

1 with 1 a b a b + + =

{0,1 }

Deutsch

Quantum Algorithms

• 1. [Feynman] Simulation of Quantum Physical Systems

with HUGE Hilber space ( 2N-D ) e.g. Strongly Correlated Electron Systems

• 2. [Peter Shor] Factoring large integers, period finding

tq ∝ Pol (N) tcl ∼ Exp (N1/3)

• 3. [Grover] Searching

tq ∝√N 〈tcl〉 ~ N/2

Digital Computer

Digital Computers in parallel

Quantum Computer : Quantum Parallelism

N bits N bits N bits N bits N bits N bits

2N

m N ⋅

1 N ⋅

{

m

Quantum Gates

Time-dependent Schrödinger Eq.

/

( ) (0) ( ) (0)

iHt

i H t t e U t ψ ψ ψ ψ ψ

−

∂ = ∂ = =

h

h ( ) ( )

1 1 1 ( ) = 1 1 1 1 1 1 1 =P( )= = = 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

i

P I X e Z Y XZ H H H

θ

θ π       = =               −       =       − −            = = = +      −           = = = −      − −      Unitary Transform Norm Preserving Reversible Hadamard

Hadamard Gate

( ) ( ) ( )

( )

1

1 2 1 2 1 1 2 2 1 2 1 2 1 2 2

... ... 1 1 1 1 1 ... 1 2 2 2 1 0 0 ...0 1 0 ...0 ... 11 ...1 2 1 ( ) 2

N

N N N N N N N N N k

H H H k

−

=

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ = + ⊗ + ⊗ + = + + + =

∑

b i n a r y e x p r e s s i

• n

Universal Quantum Gates

Xc: CNOT (controlled - NOT) or XOR

1 1 1 1 1 1 0 0 1 1

c

I X X a b a a b         ⊗ + ⊗                 = ⊗ + ⊗ = ⊕

cos sin 2 2 ( , ) sin cos 2 2

i i

ie V ie

φ φ

θ θ θ φ θ θ

− +

      −             =      −            

General Rotation of a Single Qubit

V

Quantum Circuit/Network

t

Initial State Unitary evolution : Deterministic : Reversible Universal Gates

H X

M M M

Quantum measurement : Probabilistic : Irreversible change

|Ψ〉 |0〉 |0〉 |0〉 …X1H2

CX13|Ψ〉

Scalable Qubits Cohere, Not Decohere

DiVincenzo, Qu-Ph/0002077

Quantum Key Distribution [BB84,B92] Single-Qubit Gates QKD[E91] Quantum Repeater Quantum Teleportation Single- & Two-Qubit Gates Quantum Error Correction Quantum Computer Single- & Two-Qubit Gates

40 100 ≥ ≥ 3

• 3
• √

√

7-Qubit QC√

√

Physical systems actively considered for quantum computer implementation

• Liquid-state NMR
• NMR spin lattices
• Linear ion-trap

spectroscopy

• Neutral-atom optical

lattices

• Cavity QED + atoms
• Linear optics with

single photons

• Nitrogen vacancies in

diamond

• Electrons on liquid He
• Small Josephson junctions

– “charge” qubits – “flux” qubits

• Spin spectroscopies,

impurities in semiconductors

• Coupled quantum dots

– Qubits: spin,charge, excitons – Exchange coupled, cavity coupled

Chuang Nature 414, 883-887 (20/ 27 Dec 2001) OR QP/ 0112176

15 3 5 = ×

Concept device: spin-resonance transistor

• R. Vrijen et al, Phys. Rev. A 62, 012306 (2000)

Ion Traps

• Couple lowest centre-of-mass modes to

internal electronic states of N ions.

Quantum Error Correcting Code

Three Bit Code

φ φ

M M

Um1m2

Encode Recover Decode channel noise

Molding a Quantum State

t

H X

M M M

|Ψ〉 |0〉 |0〉 |0〉 …X1H2

CX13|Ψ〉

Molding

Sculpturing a Quantum State

• Cluster state quantum computing –
• One-way quantum computing --

|+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉

1. Initialize each qubit in |+〉 state. 2. Contolled-Phase between the neighboring qubits. 3. Single qubit manipulations and single qubit measurements only [Sculpturing]. No two qubit operations!

Controlled-NOT 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AB B B A A A A A B A B AB AB AB

X I X = ⊗ + ⊗         = ⊗ + ⊗                             = +                   =       X

≡

XOR

0000 0101 1011 1110

Qubit Copying Circuit?

x x x

x x x y ⊕ y

1 a b ψ = + 1 a b ψ = + 1 a b ψ = + 00 11 a b +

Entanglement by Two-Qubit Gates

|0〉 a |0〉 + b |1〉 a |0〉|0〉 + b |1〉|1〉 |0〉 a |0〉 + b |1〉 a |0〉|0〉|0〉 + b |1〉|1〉|1〉 |0〉

Single Particle Entanglement

( )

1 1 0 0 1 2 φ = +

① ②

Beam Splitter Single Photon

① ②

Single Electron

( )

( )

( )

1 2 2 1 1 2 1 2

1 1 2 2 1

• r

f e e f 2 H t c c c c ψ ψ

+ +

= − + = + = +

Quantum Teleportation

using a single particle entanglement

Lee, JK, Qu-ph/0007106;

• Phys. Rev. A 63, 012305 (2001)

No Cloning Theorem

An Unknown Quantum State Cannot Be Cloned.

( ) ( ) ( ) ( ) ( )

1 Let . 2 1 Then 2 U U U α α α β β β α β γ α β γ α α β β γ γ = = ≠ = + = + ≠

<Proof>

Zurek, Wootters

If an unknow quantum state can be cloned …

• Quantum States can be measured as

accurately as possible ??? |ψ〉 ⇒ |ψ〉 , |ψ〉 , |ψ〉 , |ψ〉 … measure, measure, …

• Commnunication Faster Than Light?

|ψ〉 = |0〉A |1〉B - |1〉A |0〉B for “0” = |+〉A |-〉B - |-〉A |+〉B for “1”

“If ”

Communication Faster Than Light?

“If”an unknown quantum state can be copied;

• Alice wants to send Bob “1.”

Alice mesures her qubit in {|+>,|->}.

• Alice’s state will become |+> or |->.
• Bob’s state will become |-> or |+>.

Let’s assume it is |+>.

• Bob makes many copies of this.

He measures them in {|+>, |->}, and gets 100% |+>. He measures them in {|0>, |1>}, and gets 50% |0> and 50% |1>. Thus Bob can conclude that Alice measured her state in {|+>, |->}.

|ψ〉 = |0〉A |1〉B - |1〉A |0〉B for “0” = |+〉A |-〉B - |-〉A |+〉B for “1” |ψ〉 = |+〉A |-〉B - |-〉A |+〉B for “1”

|+〉 |+〉, |+〉, |+〉, |+〉, …

Mysterious Connection Between QM & Relativity

• Weinberg: Can QM be nonlinear?
• Experiments: Not so positive result.
• Polchinski, Gisin:

If QM is nonlinear, communication faster than light is possible.

Cluster State

• 1. Qubits on lattice sites are initialized to be |+〉.
• 2. Operate Control-Z on neighboring qubits.

1 2 H + + = =

0 0 1 1 1 1 1 1 1 1 1 1 1 1 = ⊗ + ⊗         = ⊗ + ⊗                        −   =    I X Z Cont -

1 2 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

z z i z i i i z

e e e e

π

σ σ σ σ

−

      = =       = + − +         = ⊗ + ⊗ − ⊗         − − − −                  + − −     − + − =   − + + −   − − − −    −  

• H

I I I

• nt - Z

I C

4

4

i

e

π

    =         −   =

H

Cont - Z

|0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉 |0〉

|+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉

|+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉

One-way quantum computing

|+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉

Optical Lattice, Quantum Dots, Superconductors, etc. Single qubit manipulations and single qubit measurements

• nly [Sculpturing].

No two qubit operations!

Quantum computing with

quantum-dot cellular automata

Géza Tóth and Craig S. Lent PRA 63, 052315

1-D array

Some comments

• 1-D cluster state quantum computing can be

efficiently simulated by digital computing.

• 2-D cluster state quantum computing is equivalent

to quantum circuit quantum computing, but needs more(just polynomially more) qubits.

“Proposal”: 2-D array of vertical QDs

• 2-D cluster state does not seem to be a ground

state of some Hamiltonian.

• Once the cluster state is prepared,
• nly single qubit measurements are needed.

QUIPU

• 2. A device consisting of a cord with knotted

strings of various colors attached, used by the ancient Peruvians for recording events, keeping accounts, etc.

• 1. Quantum Information Processing Unit [JK]

Computation Communication

[kí:pu]

* Peruvian Information Processing Unit for Computation and Communication * Rosary * Buddhist Beads